Geometrical frustration covers situations where a certain type of local order, favoured by physical interactions, cannot propagate throughout space. A classical example is that of pentagonal, or icosahedral, order which appears in the three-dimensional sphere packing problem. This symmetry, which is not compatible with translations, is nevertheless met, often imperfectly, in numerous materials. Such strong contradictions between local and global configurations are found in various physical systems, with different kinds of interactions, and coherence sizes. The concept of frustration then applies to metallic alloys at the atomic scale, to liquid crystal organizations like amphiphiles films or cholesteric blue phases at the scale of hundreds of thousands of ångströms, and even to larger scales with some biological structures.

Finite clusters

There is much experimental evidence that atomic finite clusters do not present necessarily the same geometrical order as in their crystalline counterparts. For instance, rare gas clusters often display icosahedral symmetry, their detailed structure depending on atomic species and size (Farge et al. 1975). It is now possible to grow and study metallic or covalent clusters, ranging from a few atoms up to several hundred atoms (Joyes 1990, Paillard et al. 1994). They are interesting not only for themselves (in the context of catalysis for example) but also to better understand cohesive properties of solids, or in the context of amorphous structures, since they can possibly model the local order of the non-crystalline solid during the growth process. One should not forget however that surface effects play a dominant role in cluster stability while their role in glasses should only be invoked for a dynamical process.

Several types of cluster have been proposed in the past. Some of them have a close relationship with mapped polytopes. The main results have been summarized elsewhere (Mosseri 1988a, Mosseri and Sadoc 1989), which we now describe.

Cluster indexation

We want to index finite clusters which can be derived from a given polytope P whose sites are decorated by atoms. A finite portion of P is mapped onto a tangent hyperplane R3. We need to specify the tangent point T and the polar angle ω which limits the region to be mapped, and as a consequence the size of the mapped cluster.

Frank and Kasper polyhedra

One can always, in an unambiguous way, divide up a structure, made of points, into (usually irregular) tetrahedra. This is done using first the Voronoi (or Dirichlet) decomposition of space into individual cells which contain the regions of space, closer to a given point than to any other one. In generic cases, the Voronoi cells have three faces sharing a vertex of the cell. Then, connecting the original points of the set whenever their associated Voronoi cells share a face, defines a unique decomposition of the space into tetrahedra. This simplicial decomposition is equivalent, in three dimensions, to a point set triangulation in two dimensions. This procedure also provides the best way to define the coordination number in dense structure: it is the number of faces of the Voronoi cell. In a topological sense the Voronoi cell and the coordination polyhedra are dual. In a tetrahedral division of space, the set of vertices closest to a given site form its first coordination shell, which is a triangulated polyhedron (a deltahedron).

Let us introduce now a standard notation for a site coordination in a tetrahedrally close-packed structure. If the tetrahedra are not too distorted, we can only find situations where either five or six tetrahedra share a given edge. This is the case considered by Frank and Kasper (1958), who then proposed the following notation: a site such that its first neighbour shell is an icosahedron (allowing for small distortions) is called a Z12 site.

Hierarchical polytopes and symmetry groups

Symmetry group of the {3, 3, 5} polytope and homotopy theory of defects

Line defects in condensed media can be classified using the homotopy theory (Toulouse and Kléman 1976), which allows us to identify the possible defects and to analyse their mutual relationships (Rivier and Duffy 1982). In the polytope {3, 3, 5}, linear defects are given by the conjugacy classes of the fundamental group π1(SO(4)/G′) where G′ is the direct symmetry group of the polytope (Nelson and Widom 1984).

The polytope full symmetry group is presented in appendix A2, §3 using quaternions and in appendix 4 using the orthoscheme description. It contains 14 400 elements while the direct group G′ contains 7200 elements. It is related to the icosahedral group Y by: G′ = Y′ × Y′ / Z2, where Z2 is a two element group and Y′, of order 120, is the lift of the R3 icosahedral group Y into SU(2) (SU(2) is isomorphic to the group of unit quaternions). This shows that elements of G′ can be described as a combination of screws (sometimes called ‘double rotations’). Related defects also combine several screws along different axes, leading to ‘dispirations’ similar to those introduced by Harris (see §4.1).

Among these defects are those related to 2π/5 rotations. The introduction of two interlaced disclinations of this kind into the polytope {3, 3, 5} has already been presented in chapter 4.

Frustration and large cell crystals

We have described the occurrence of structural complexity in non-periodic systems as the result of geometrical frustration. This applies naturally to non-periodic structures like amorphous structures or quasicrystals, but also to some crystalline structures. In particular, it is interesting and instructive to analyse those periodic structures which also contain ingredients of frustration as a result, for instance, of local icosahedral configurations. We shall mainly focus on metals, or on clathrate structures which are dual from a geometrical point of view. Note that similar considerations could prove useful in systems with non-metallic interactions, for instance the C60 molecular crystal or the boron structure.

A major result of the above curved space approach is to propose a description of structural complexity at a range which is larger than that defined by the frustrated local interactions. Once the ideal structure has been defined in curved space, the appropriate range is related to the radius of curvature of the latter, which governs the average distance between the disclination decurving defects. The complexity is then encoded in the defect network itself. If ever this network adopts a periodic structure, this will often give rise to a large cell crystal. Since the paradigmatic example of frustration is to be found in close-packed metallic systems, it is then natural to look for their possible occurrence in nature. We shall also look at tetracoordinated (§7.4) and liquid crystalline (§7.5) periodic structures, in which similar networks can be recognized.

Introduction

Any subgroup of the full rotation group in three dimensions O(3, r) (= O(3)) is called a point group, because all the group elements leave at least one point (the coordinate origin) in space invariant. Since there are a great deal of applications for the point groups of finite order, we shall discuss their group structures in greater detail. There exist five types of proper and nine types of improper point groups of finite order. A new system of notations for the latter is introduced by expressing an improper operation with the inversion or a rotation–inversion. This system is very effective in describing the isomorphisms between proper and improper point groups, because the inversion commutes with any point operation.

Following the historical development, we shall introduce a point group by the symmetry point group of a geometric body such as a regular polyhedron. The symmetry of a geometric body is defined by the set of all symmetry transformations which brings the body into coincidence with itself. By definition, such a set of transformations forms a group, the symmetry group G of the body. More specifically, let p be a point on the body, then a set of transformations which leaves the point p invariant forms a subgroup H of G.

As has been shown in Section 9.4, the parameter space of the proper rotation group SO(3, r) is doubly connected so that there exist single-valued as well as double-valued representations for SO(3, r). In this chapter, it will be shown that the special unitary group SU(2) is simply connected and homomorphic to SO(3, r) with two-to-one correspondence. Accordingly, the representations of SU(2) are single-valued, but these provide all the single-and double-valued representations of SO(3, r). In particular, SU(2) itself provides a 2 × 2 matrix representation of SO(3, r) that is double-valued. As was mentioned at the end of Section 4.1, an element of SU(2) is called a spinor transformation, because of the role it plays in the theory of the spinning electron. Moreover, the double-valued unirreps of SO(3, r) given by the unirreps of SU(2) are called the spinor representations of SO(3, r). We shall begin with a discussion on the special unitary group SU(2).

The structure of SU(2)

The generators of SU(2)

Previously, in Section 4.1, we have shown that an element of the special unitary group in n dimensions SU(n) is expressed by a matrix of the form U0 = exp K0, where K0 is an anti-Hermitian traceless matrix.

Introduction

An infinite group is a group that contains an infinite number of elements. The group axioms still hold for infinite groups. Among infinite groups, there are two categories: discrete and continuous ones. If the number of elements of a group is denumerably infinite, the group is said to be discrete, whereas if the number of elements is nondenumerably infinite, it is called a continuous group. For example, the whole set of rational numbers forms an infinite group that is discrete, whereas the whole set of real positive numbers is a continuous group. A continuous group G is a set of group elements that can be characterized by a set of continuous real parameters in a certain region called the parameter domain (or space) Ω such that there exists a one-to-one correspondence between group elements in G and points (the parameter sets) in the parameter domain Ω. For example, an element of the rotation group SO(3, r) = {R(θ)} is characterized by a set of three real parameters θ = (θ1, θ2, θ3) in the parameter sphere Ω of the radius π, i.e. 0 ≤ |θ| ≤ π with the cyclic boundary condition (see Equation (4.3.6)). In a continuous group G, the nearness of group elements is characterized by the nearness of their parameters in Ω. Thus the neighborhood of a group element is characterized by the neighborhood of the corresponding parameter set.